How was born the topology from Euclidean geometry? Good evening, I have the question arose as to create interest starting topology of Euclidean geometry, what was the interest of those who created it?
Thanks for your help
 A: You have to distinguish between general topology and algebraic topology.  For general topology, a key idea was to define "closeness" for Riemann surfaces, in order to give complex functions such as  $\surd z, \log z$ a single value by defining them on something lying "over" part of the usual complex plane $\mathbb C$. This led to Hausdorff's notion of "neighbourhood space" and then to the logically simpler, but less intuitive,  notion of a topology defined by open sets. 
The origin of algebraic topology lies first with Euler's solution of the Konigsberg Bridge problem, and then later with workers defining Betti numbers and torsion coefficients of higher dimensional spaces. This led to the problem of defining notions such as "boundary" and "cycles" so that one could think of Betti numbers as arising from "cycles modulo boundaries". Poincare in his great series of papers on "Analysis Situs", later called "Topology", solved this problem by taking the free abelian group on the oriented simplices of a given dimension $n$ and defining the boundary $\partial$ of an $n$-simplex $\sigma$ by 
$$\partial \sigma= \sum_\rho \epsilon(\sigma,\rho] \rho  $$
for $\rho$ of dimension $n-1$ in the boundary of $\sigma$, and $\epsilon$ denotes the chosen orientations. This was much later simplified by Eilenberg by choosing an ordering of all the vertices so that one could define 
$$ \partial \sigma = \sum _i (-1)^i \partial _i \sigma $$
leading to the crucial rule $\partial \partial =0$, so that every boundary $z= \partial \sigma$ was a cycle, i.e. $\partial z = 0$. 
A: Topology was a convergence of ideas from (at least) geometry, graph theory, and analysis. See Mactutor. Basically, different people were working on different problems in different fields and ran into commonalities that eventually came to be called topology.
A: As far as I know, the main interest was to find a new way of thinking about the concept of two points being 'close' to each other without using the idea of Euclidean distance because many ideas about convergence of sequences, series, limits and other things can be thought in a broader and deeper context.
Back then in the time of Cauchy, people were busy with creating a reasonable approach to Calculus that is not based on infinitesimals. The definition of limit which is based on $\epsilon-\delta$ definition comes from that time. But the $\epsilon-\delta$ definition still uses an important concept that is not as general as we want, namely, the Euclidean metric. So, people started to study functions that they wanted them to satisfy some intuitive properties that came from their experience in geometry and called them metric functions.But later it was realized that in some new branches of mathematics like graph theory the notion of 'distance' provided by a metric is unnecessary and later mathematicians realized that most notions like limits can be studied in a broader ground if we think about 'open' sets. Then mathematicians tried to find an axiomatic way to create a new science which tries to explain the concept of 'closeness' using something abstract like the notion of an 'open' set. I personally know of four equivalent axioms for defining a topology. One is related to open sets, the other is related to close sets which are defined as the compliment of open sets, there's another beautiful definition based on neighborhoods where neighborhoods are thought of abstractly, and the other one that I know of is due to Kuratowski, a famous Polish topologist, and uses closures.
